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Lim x tending to 0 ( 1/e^2x -1 -1/ax) Exists and equal to l then find the value of a and l ?
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Lim x tending to 0 ( 1/e^2x -1 -1/ax) Exists and equal to l then find ...
Solution:
To find the value of 'a' and 'l', we need to evaluate the limit as x approaches 0 for the given expression:
lim(x→0) [1/e^(2x) - 1 - 1/(ax)]
We will simplify the expression step by step to evaluate the limit.
Simplifying the expression:
Step 1: Rewrite the expression using the properties of exponents.
lim(x→0) [1/(e^(2x)) - 1 - 1/(ax)]
Step 2: Find a common denominator for the terms.
The common denominator is ax * e^(2x).
lim(x→0) [(ax - e^(2x) - e^(2x))/(ax * e^(2x))]
Step 3: Combine the terms in the numerator.
lim(x→0) [(ax - 2e^(2x))/(ax * e^(2x))]
Taking out the common factor:
Step 4: Factor out 'e^(2x)' from the numerator.
lim(x→0) [e^(2x)(a - 2)/(ax * e^(2x))]
Step 5: Cancel out 'e^(2x)' from the numerator and denominator.
lim(x→0) (a - 2)/(ax)
Evaluating the limit:
Step 6: Substitute x = 0 into the expression.
(a - 2)/(a * 0)
Step 7: Simplify the expression.
This expression is indeterminate as it results in a division by zero.
To find the value of 'a' and 'l', we need to evaluate the limit as x approaches 0 for the given expression:
lim(x→0) [1/e^(2x) - 1 - 1/(ax)]
We will simplify the expression step by step to evaluate the limit.
Simplifying the expression:
Step 1: Rewrite the expression using the properties of exponents.
lim(x→0) [1/(e^(2x)) - 1 - 1/(ax)]
Step 2: Find a common denominator for the terms.
The common denominator is ax * e^(2x).
lim(x→0) [(ax - e^(2x) - e^(2x))/(ax * e^(2x))]
Step 3: Combine the terms in the numerator.
lim(x→0) [(ax - 2e^(2x))/(ax * e^(2x))]
Taking out the common factor:
Step 4: Factor out 'e^(2x)' from the numerator.
lim(x→0) [e^(2x)(a - 2)/(ax * e^(2x))]
Step 5: Cancel out 'e^(2x)' from the numerator and denominator.
lim(x→0) (a - 2)/(ax)
Evaluating the limit:
Step 6: Substitute x = 0 into the expression.
(a - 2)/(a * 0)
Step 7: Simplify the expression.
This expression is indeterminate as it results in a division by zero.
Therefore, the given limit does not exist for any value of 'a'.
To find the value of 'a' and 'l', we need to evaluate the limit as x approaches 0 for the given expression:
lim(x→0) [1/e^(2x) - 1 - 1/(ax)]
We will simplify the expression step by step to evaluate the limit.
Simplifying the expression:
Step 1: Rewrite the expression using the properties of exponents.
lim(x→0) [1/(e^(2x)) - 1 - 1/(ax)]
Step 2: Find a common denominator for the terms.
The common denominator is ax * e^(2x).
lim(x→0) [(ax - e^(2x) - e^(2x))/(ax * e^(2x))]
Step 3: Combine the terms in the numerator.
lim(x→0) [(ax - 2e^(2x))/(ax * e^(2x))]
Taking out the common factor:
Step 4: Factor out 'e^(2x)' from the numerator.
lim(x→0) [e^(2x)(a - 2)/(ax * e^(2x))]
Step 5: Cancel out 'e^(2x)' from the numerator and denominator.
lim(x→0) (a - 2)/(ax)
Evaluating the limit:
Step 6: Substitute x = 0 into the expression.
(a - 2)/(a * 0)
Step 7: Simplify the expression.
This expression is indeterminate as it results in a division by zero.
To find the value of 'a' and 'l', we need to evaluate the limit as x approaches 0 for the given expression:
lim(x→0) [1/e^(2x) - 1 - 1/(ax)]
We will simplify the expression step by step to evaluate the limit.
Simplifying the expression:
Step 1: Rewrite the expression using the properties of exponents.
lim(x→0) [1/(e^(2x)) - 1 - 1/(ax)]
Step 2: Find a common denominator for the terms.
The common denominator is ax * e^(2x).
lim(x→0) [(ax - e^(2x) - e^(2x))/(ax * e^(2x))]
Step 3: Combine the terms in the numerator.
lim(x→0) [(ax - 2e^(2x))/(ax * e^(2x))]
Taking out the common factor:
Step 4: Factor out 'e^(2x)' from the numerator.
lim(x→0) [e^(2x)(a - 2)/(ax * e^(2x))]
Step 5: Cancel out 'e^(2x)' from the numerator and denominator.
lim(x→0) (a - 2)/(ax)
Evaluating the limit:
Step 6: Substitute x = 0 into the expression.
(a - 2)/(a * 0)
Step 7: Simplify the expression.
This expression is indeterminate as it results in a division by zero.
Therefore, the given limit does not exist for any value of 'a'.
Community Answer
Lim x tending to 0 ( 1/e^2x -1 -1/ax) Exists and equal to l then find ...
To find the value of "a" and "l" in the given expression, let's first simplify the expression and then apply the limit.
Simplifying the expression:
1/e^(2x) - 1 - 1/(ax) = (1 - e^(2x))/(e^(2x)) - 1/(ax)
Now, let's find the limit as x tends to 0:
lim(x→0) ((1 - e^(2x))/(e^(2x))) - (1/(ax))
To evaluate this limit, we can apply L'Hôpital's rule. This rule states that if we have an indeterminate form 0/0 or ∞/∞, we can differentiate the numerator and denominator separately until we no longer have an indeterminate form.
Differentiating the numerator and denominator:
lim(x→0) (d/dx((1 - e^(2x))/(e^(2x)))) - (d/dx(1/(ax)))
Differentiating the numerator:
lim(x→0) (d/dx(1 - e^(2x))) / (d/dx(e^(2x)))
Differentiating with respect to x:
lim(x→0) (-2e^(2x)) / (2e^(2x))
Simplifying:
lim(x→0) -e^(2x) / e^(2x) = -1
Therefore, the value of "l" is -1.
Now, let's find the value of "a". We can substitute the value of "l" into the original expression and solve for "a".
-1 = 1/e^(2x) - 1 - 1/(ax)
Multiplying through by e^(2x):
-e^(2x) = 1 - e^(2x) - e^(2x)/(ax)
Simplifying:
0 = 1 - 2e^(2x) - e^(2x)/(ax)
Rearranging the terms:
2e^(2x) = 1 - e^(2x) + e^(2x)/(ax)
Combining like terms:
2e^(2x) = 1 + e^(2x)/(ax)
Multiplying through by ax:
2axe^(2x) = ax + e^(2x)
Bringing all terms to one side:
2axe^(2x) - ax - e^(2x) = 0
Factoring out common terms:
(ax - 1)(2xe^(2x) - 1) = 0
Setting each factor equal to zero:
ax - 1 = 0 or 2xe^(2x) - 1 = 0
For ax - 1 = 0:
ax = 1
a = 1/x
For 2xe^(2x) - 1 = 0:
2xe^(2x) = 1
xe^(2x) = 1/2
x = (1/2e^(2x))
Therefore, the value of "a" is 1/x, where x = (1/2e^(2x)).
Note: The value of "a" cannot be determined uniquely without more information or constraints on the problem.
Simplifying the expression:
1/e^(2x) - 1 - 1/(ax) = (1 - e^(2x))/(e^(2x)) - 1/(ax)
Now, let's find the limit as x tends to 0:
lim(x→0) ((1 - e^(2x))/(e^(2x))) - (1/(ax))
To evaluate this limit, we can apply L'Hôpital's rule. This rule states that if we have an indeterminate form 0/0 or ∞/∞, we can differentiate the numerator and denominator separately until we no longer have an indeterminate form.
Differentiating the numerator and denominator:
lim(x→0) (d/dx((1 - e^(2x))/(e^(2x)))) - (d/dx(1/(ax)))
Differentiating the numerator:
lim(x→0) (d/dx(1 - e^(2x))) / (d/dx(e^(2x)))
Differentiating with respect to x:
lim(x→0) (-2e^(2x)) / (2e^(2x))
Simplifying:
lim(x→0) -e^(2x) / e^(2x) = -1
Therefore, the value of "l" is -1.
Now, let's find the value of "a". We can substitute the value of "l" into the original expression and solve for "a".
-1 = 1/e^(2x) - 1 - 1/(ax)
Multiplying through by e^(2x):
-e^(2x) = 1 - e^(2x) - e^(2x)/(ax)
Simplifying:
0 = 1 - 2e^(2x) - e^(2x)/(ax)
Rearranging the terms:
2e^(2x) = 1 - e^(2x) + e^(2x)/(ax)
Combining like terms:
2e^(2x) = 1 + e^(2x)/(ax)
Multiplying through by ax:
2axe^(2x) = ax + e^(2x)
Bringing all terms to one side:
2axe^(2x) - ax - e^(2x) = 0
Factoring out common terms:
(ax - 1)(2xe^(2x) - 1) = 0
Setting each factor equal to zero:
ax - 1 = 0 or 2xe^(2x) - 1 = 0
For ax - 1 = 0:
ax = 1
a = 1/x
For 2xe^(2x) - 1 = 0:
2xe^(2x) = 1
xe^(2x) = 1/2
x = (1/2e^(2x))
Therefore, the value of "a" is 1/x, where x = (1/2e^(2x)).
Note: The value of "a" cannot be determined uniquely without more information or constraints on the problem.
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Lim x tending to 0 ( 1/e^2x -1 -1/ax) Exists and equal to l then find the value of a and l ? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Lim x tending to 0 ( 1/e^2x -1 -1/ax) Exists and equal to l then find the value of a and l ? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Lim x tending to 0 ( 1/e^2x -1 -1/ax) Exists and equal to l then find the value of a and l ?.
Lim x tending to 0 ( 1/e^2x -1 -1/ax) Exists and equal to l then find the value of a and l ? for Mathematics 2025 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Lim x tending to 0 ( 1/e^2x -1 -1/ax) Exists and equal to l then find the value of a and l ? covers all topics & solutions for Mathematics 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Lim x tending to 0 ( 1/e^2x -1 -1/ax) Exists and equal to l then find the value of a and l ?.
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